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Abstract In this paper, we consider an initial-boundary value problem of the p-Laplacian parabolic equation { u t (x , t) = div (| ∇ u (x , t) | p − 2 ∇ u (x , t)) + f (u (x , t)) , (x , t) ∈ Ω × (0 , + ∞) , u (x , t) = 0 , (x , t) ∈ ∂ Ω × [ 0 , + ∞) , u (x , 0) = u 0 ≥ 0 , x ∈ Ω ‾ , where p ≥ 2 and Ω is a bounded domain of R N (N ≥ 1) with smooth boundary ∂Ω. The main contribution of this work is to introduce a new condition (C p) α ∫ 0 u f (s) d s ≤ u f (u) + β u p + γ , u > 0 for some α , β , γ > 0 with 0 < β ≤ (α − p) λ 1 , p p , where λ 1 , p is the first eigenvalue of p-Laplacian Δ p , and we use the concavity method to obtain the blow-up solutions to the above equations. In fact, it will be seen that the condition (C p) improves the conditions ever known so far. [ABSTRACT FROM AUTHOR]

Abstract: In this paper we study the stability of the nonlinear wave structure caused by the attack of an incident shock on an interface of two different kinds of media. The attack will produce a reflected wave and a refracted wave, and also let the interface deflected. In this paper we will mainly study the case, when the reflected wave is a shock, and the flow between the reflected wave and the refracted shock is relatively subsonic. Our result indicates that the wave structure and the flow field for the reflection–refraction problem in this case is conditionally stable. To describe the motion of the fluid we use the inviscid Euler system as the mathematical model. The reflection–refraction problem can be reduced to a free boundary value problem, where the unknown reflected shock and refracted shock are free boundaries, and the deflected interface is also to be determined. In the proof of the existence and the stability of the corresponding wave structure we apply the Lagrange transformation to fix the interface and the decoupling technique to decouple the elliptic–hyperbolic composite system in its principal part. Meanwhile, some efficient weighted Sobolev estimates are established to derive the existence for corresponding nonlinear problems. [Copyright &y& Elsevier]

Abstract: In this paper, a kind of Riemann problem for the Euler equations in a van der Waals fluid is considered. We constructed the weak solution in multidimensional space which contains one shock front and one subsonic phase boundary. We mainly follow the arguments of Majda''s [A. Majda, The stability of multi-dimensional shock fronts, Mem. Amer. Math. Soc. 275 (1983) 1–95; A. Majda, The existence of multi-dimensional shock fronts, Mem. Amer. Math. Soc. 281 (1983) 1–93] and Métivier''s [G. Métivier, Interaction de deux chocs pour un système de deux lois de conservation, en dimension deux d''espace, Trans. Amer. Math. Soc. 296 (1986) 431–479] work. The linear stability results are based on Majda''s [A. Majda, The stability of multi-dimensional shock fronts, Mem. Amer. Math. Soc. 275 (1983) 1–95] work for the single shock front and Wang and Xin''s [Y.-G. Wang, Z. Xin, Stability and existence of multidimensional subsonic phase transitions, Acta Math. Appl. Sin. 19 (2003) 529–558] work for the single phase boundary. The initial boundary value problem concerned in this paper is different from the boundary value problem for double shock fronts concerned in [G. Métivier, Interaction de deux chocs pour un système de deux lois de conservation, en dimension deux d''espace, Trans. Amer. Math. Soc. 296 (1986) 431–479], we slightly modified Métivier''s frame work to establish the existence for the solution to the nonlinear problem. [Copyright &y& Elsevier]

Abstract: In this paper we prove that the spectral radius of the traction double layer potential operator associated with the Lamé system on an infinite sector in is within 10−2 from a certain conjectured value which depends explicitly on the aperture of the sector and the Lamé moduli of the system. This type of result is relevant to the spectral radius conjecture, cf., e.g., Problem 3.2.12 in [C.E. Kenig, Harmonic Analysis Techniques for Second Order Elliptic Boundary Value Problems, CBMS Reg. Conf. Ser. Math., vol. 83, Amer. Math. Soc., Providence, RI, 1994]. The techniques employed in the paper are a blend of classical tools such as Mellin transforms, and Calderón–Zygmund theory, as well as interval analysis—resulting in a computer-aided proof. [Copyright &y& Elsevier]

Abstract: In this paper we are interested in the global existence and large-time behavior of solutions to the initial-boundary value problem for subcritical Kuramoto–Sivashinsky-type equationwhere the nonlinear term depends on the unknown function u and its derivative and satisfy the estimatewith such that The aim of this paper is to prove the global existence of solutions to the initial-boundary value problem (0.1) in subcritical case, when the nonlinear term has a time decay rate less than that of the linear terms of Eq. (0.1). Also we find the main term of the asymptotic representation of solutions. [Copyright &y& Elsevier]

Abstract: In this paper, we study the existence of positive solutions and sign-changing solutions for the following boundary value problem in the half-space where λ is a positive parameter and the nonlinear term f is superlinear at zero and asymptotically linear at infinity. [ABSTRACT FROM AUTHOR]

Abstract: For a cylindrical superconductor surrounded by a normal material, we discuss transition to the normal phase of stable, locally stable and critical configurations. Associated with those phase transitions, we define critical magnetic fields and we provide a sufficient condition for which those critical fields coincide. In particular, when the conductivity ratio of the superconducting and the normal material is large, we show that the aforementioned critical magnetic fields coincide, thereby proving that the transition to the normal phase is sharp. One key-ingredient in the paper is the analysis of an elliptic boundary value problem involving ‘transmission’ boundary conditions. Another key-ingredient involves a monotonicity result (with respect to the magnetic field strength) of the first eigenvalue of a magnetic Schrödinger operator with discontinuous coefficients. [Copyright &y& Elsevier]

Abstract: We address the analysis of a nonlinear and degenerating PDE system, proposed by M. Frémond for modelling phase transitions in viscoelastic materials subject to thermal effects. The system features an internal energy balance equation, governing the evolution of the absolute temperature ϑ, an evolution equation for the phase change parameter χ, and a stress–strain relation for the displacement variable u. The main novelty of the model is that the equations for χ and u are coupled in such a way as to take into account the fact that the properties of the viscous and of the elastic parts influence the phase transition phenomenon in different ways. However, this brings about an elliptic degeneracy in the equation for u which needs to be carefully handled. In this paper, we first prove a local (in time) well-posedness result for (a suitable initial–boundary value problem for) the above mentioned PDE system, in the (spatially) three-dimensional setting. Secondly, we restrict to the one-dimensional case, in which, for the same initial–boundary value problem, we indeed obtain a global well-posedness theorem. [Copyright &y& Elsevier]

Abstract: In this paper, we establish the global attractivity of the positive steady state of the diffusive Nicholson''s equation with homogeneous Neumann boundary value under a condition that makes the equation a non-monotone dynamical system. To achieve this, we develop a novel method: combining a dynamical systems argument with maximum principle and some subtle inequalities. [Copyright &y& Elsevier]

Abstract: This paper examines the time-dependent double diffusive convection in Brinkman flow in a semi-infinite cylinder. Under appropriate initial and boundary conditions the authors establish exponential decay of solutions in “energy” norm with distance from the finite end of the cylinder. [Copyright &y& Elsevier]

Abstract: In this paper we consider the one-dimensional p-Laplacian boundary value problem on time scales where is p-Laplacian operator, i.e., , . Some new results are obtained for the existence of at least single, twin or triple positive solutions of the above problem by using Krasnosel''skii''s fixed point theorem, new fixed point theorem due to Avery and Henderson and Leggett–Williams fixed point theorem. This is probably the first time the existence of positive solutions of one-dimensional p-Laplacian boundary value problems on time scales has been studied. [Copyright &y& Elsevier]

Abstract: In this paper, some sufficient conditions for the existence of solutions to the boundary value problems of a class of second order difference equation are obtained by using the critical point theory. [Copyright &y& Elsevier]

Abstract: In this paper, we discuss an initial-boundary value problem and a Cauchy problem for the stochastic porous medium equation. Our basic estimates are based on the known results due to (Math. Sbornik 67 (1965) 609–642, Arch. Rational Mech. Anal. 25 (1967) 64–80). By the procedure developed in (Trans. Amer. Math. Soc. 354 (2001) 1117–1135), we obtain solutions over the given probability space rather than martingale solutions. We will also establish the existence of invariant measures when the space domain is bounded. [Copyright &y& Elsevier]

Abstract: In this paper, we consider the initial-boundary value problem of a semilinear parabolic equation with local and non-local (localized) reactions in a ball: in where and . If , there exist blow-up solutions of this problem for large initial data. We treat the radially symmetric and one peak non-negative solution of this problem. We give the complete classification of total blow-up phenomena and single point blow-up phenomena according to p and q. [(i)] If or , then single point blow-up occurs whenever solutions blow up. [(ii)] If , both phenomena, total blow-up and single point blow-up, occur depending on the initial data. [(iii)] If , total blow-up occurs whenever solutions blow up. [(iv)] If , every solution exists globally in time. [Copyright &y& Elsevier]

Abstract: This paper is devoted to study the following third-order multi-point singularly perturbed boundary value problemwhere , , , , and . The existence, uniqueness and asymptotic estimates of solutions of the boundary value problem are give by using priori estimates, differential inequalities technique and Leray–Schauder degree theory. [Copyright &y& Elsevier]

Abstract: In this paper we consider boundary-value problems in domains with perforated boundaries. We use the classification of homogenized (limit) problems depending on the ratio of small parameters, which characterize the diameter of the holes and the distance between them. We study the analogue of the Helmholtz resonator for domains with a perforated boundary. [Copyright &y& Elsevier]